Determine a rule from an input-output table. - ISEE Lower Level Quantitative Reasoning

SECTION 1

Where Did Input-Output Tables Come From?

People have always loved finding patterns! Thousands of years ago, farmers noticed patterns in the seasons. Traders noticed patterns in prices. Mathematicians started writing these patterns down in tables so they could predict what would happen next.

An input-output table is a simple chart with two columns. You put a number in (the input), a secret rule changes it, and a new number comes out (the output). Let's see how this idea grew over time!

2000 BC • Ancient Babylon

Babylonian scribes carved multiplication tables into clay tablets. These were some of the first input-output tables ever made!

300 BC • Ancient Greece

Greek thinkers like Euclid studied number patterns. They loved finding rules that connected one number to another.

1600s • The Idea of Functions

Mathematicians invented the word "function" to describe a rule that turns an input into an output. This is exactly what you'll learn today!

Today • ISEE & School Math

Input-output tables appear on the ISEE and in classrooms everywhere. Finding the rule is like cracking a secret code!

So here's the big question: if someone gives you a table of numbers, can you figure out the hidden rule? That's exactly what this lesson teaches. Let's go!

SECTION 2

Core Ideas You Need to Know

Before we start cracking codes, let's learn a few important words. These ideas will pop up again and again on the ISEE!

Input

The number you start with. It goes INTO the rule. You'll see it in the left column of the table.

Output

The number that comes out after the rule is applied. It appears in the right column of the table.

Rule (Function)

The hidden operation that changes every input into its output. It could be adding, subtracting, multiplying, or dividing.

Consistency

The same rule works for every row in the table. If your guess doesn't work for all rows, keep trying!

✦

KEY TAKEAWAY

Think of an input-output table like a magic number machine. You drop a number in the top, the machine does something secret to it, and a new number pops out the bottom. Your job is to figure out what the machine does!

SECTION 3

See the Machine in Action

Let's picture our magic number machine. Look at the diagram below. Each input goes in on the left, the secret rule is applied inside the machine, and the output appears on the right.

Each blue input enters the machine. The secret rule (× 3) transforms it into a pink output. Notice how the same rule works for every row!

See how 2 goes in and 6 comes out? That's because 2 × 3 = 6. The number 5 goes in and 15 comes out because 5 × 3 = 15. The same rule works every single time. That's how you know it's the right one!

SECTION 4

How to Find the Rule — Step by Step

Here's a simple strategy you can use on the ISEE every time you see an input-output table. Follow these steps and you'll find the rule!

THE GENERAL IDEA

Input → Rule → Output

The rule is what you do to the input to get the output. It could be + , − , × , or ÷ by a number.

ADDITION RULE EXAMPLE

Input + 5 = Output

If Input = 3 and Output = 8, you can check: 3 + 5 = 8. ✓

MULTIPLICATION RULE EXAMPLE

Input × 4 = Output

If Input = 6 and Output = 24, you can check: 6 × 4 = 24. ✓

💡

ISEE TEST TIP

Always check your rule with every row in the table, not just one. If it works for all rows, you've found it! If it fails even once, try a different operation.

Step 1: Look at the first row. Ask: what can I do to the input to get the output?
Step 2: Try adding, subtracting, multiplying, or dividing.
Step 3: Test your guess on the second and third rows.
Step 4: If it works for every row, that's your rule! Pick that answer.

SECTION 5

Common Rule Types on the ISEE

On the ISEE, the rules usually involve one operation. Here are the four types you'll see most often. Let's look at each one so you know what to expect!

The four main rule types you'll see on the ISEE. Notice: with addition the output is slightly bigger, with multiplication it's much bigger, with subtraction it's smaller, and with division it's much smaller.

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QUICK CLUE

Compare the size of the output to the input. If the output is bigger, try adding or multiplying first. If the output is smaller, try subtracting or dividing. This saves time on the test!

SECTION 6

Let's Solve One Together!

Here's a problem just like one you might see on the ISEE. Let's work through it step by step.

Input	Output
4	12
7	21
9	27
11	?

What is the rule? What is the missing output?

Finding the Rule

1
Step 1 — Compare the First Row

The input is 4 and the output is 12. The output is bigger, so let's try addition or multiplication. Is 4 + something = 12? Yes, 4 + 8 = 12. But let's also check: is 4 × something = 12? Yes, 4 × 3 = 12. We have two guesses!

Possible rules: + 8 or × 3
2
Step 2 — Test Both Guesses on the Second Row

The second input is 7 and the output is 21. Let's test + 8 first: 7 + 8 = 15. But the output is 21, not 15. So + 8 is wrong! Now test × 3: 7 × 3 = 21. That matches!

Rule × 3 works! Rule + 8 is eliminated.
3
Step 3 — Confirm with the Third Row

The third input is 9. Using our rule × 3: 9 × 3 = 27. The table says 27. Perfect — it matches!

Confirmed! The rule is × 3.
4
Step 4 — Find the Missing Output

Now use the rule on the last row. The input is 11. Apply the rule: 11 × 3 = 33.

The missing output is 33.

🎯

REMEMBER THIS STRATEGY

When two guesses both work for one row, don't panic! Just test them on the next row. The wrong guess will fail, and the right one will survive. It's like a detective eliminating suspects!

SECTION 7

Tips, Tricks & Common Mistakes

Now you know the steps. Let's talk about what helps and what trips students up. Knowing these tips before test day gives you a big advantage!

Helpful Tips ✅	Common Mistakes ❌
Always check your rule on EVERY row.	Checking only one row and guessing too fast.
Compare size: bigger output → try + or ×.	Trying subtraction when the output is larger.
If the outputs are much bigger, try × first.	Confusing + and × (e.g., + 8 vs. × 2).
Use process of elimination on the answer choices.	Not reading all four answer choices before picking.
Never leave a question blank — there's no penalty for guessing!	Leaving the answer blank when unsure.

⚡

ISEE STRATEGY

On the ISEE, there's no penalty for a wrong answer. If you're stuck, cross out answers that don't work for the first row and guess from what's left. You've got nothing to lose!

SECTION 8

From Simple Rules to Bigger Ideas

You're building an awesome skill right now! The input-output tables on the ISEE Lower Level use one operation. As you grow in math, you'll see tables with two steps (like "multiply by 2, then add 1"). For now, let's see how what you're learning connects to bigger math ideas.

What You Learn Now	What Comes Later
One-step rules (+ 5, × 3)	Two-step rules (× 2 + 1)
Input-output tables	Equations with variables like y = 3x
Finding patterns in numbers	Graphing lines on a coordinate plane
Checking the rule on every row	Proving formulas in algebra

Everything you learn about finding rules today is the foundation for algebra later. You're getting a head start! Great job!

SECTION 9

Practice Problems

Time to put your skills to the test! These problems start easy and get harder. Remember: check every row, and never leave a question blank. You've got this!

Question 1: CONCEPTUAL

Look at this table. Input: 2 → Output: 7 Input: 5 → Output: 10 Input: 9 → Output: 14 What is the rule?

A. Add 3
B. Add 5
C. Multiply by 2
D. Multiply by 3

Show answer (B)

The correct answer is B (Add 5). Check: 2 + 5 = 7 ✓, 5 + 5 = 10 ✓, 9 + 5 = 14 ✓. It works for all rows! A (Add 3) is wrong because 2 + 3 = 5, not 7. C (Multiply by 2) is wrong because 2 × 2 = 4, not 7. D (Multiply by 3) is wrong because 2 × 3 = 6, not 7.

Question 2: BASIC CALCULATION

Look at this table. Input: 3 → Output: 12 Input: 6 → Output: 24 Input: 8 → Output: 32 If the input is 10, what is the output?

A. 30
B. 36
C. 40
D. 44

Show answer (C)

The correct answer is C (40). The rule is × 4. Check: 3 × 4 = 12 ✓, 6 × 4 = 24 ✓, 8 × 4 = 32 ✓. So 10 × 4 = 40. A (30) would come from × 3, but 3 × 3 = 9, not 12. B (36) would come from + 26, which doesn't match other rows. D (44) would come from + 34, which also doesn't match.

Question 3: INTERMEDIATE

Look at this table. Input: 20 → Output: 15 Input: 16 → Output: 11 Input: 30 → Output: 25 What is the rule?

A. Subtract 3
B. Subtract 5
C. Divide by 2
D. Subtract 10

Show answer (B)

The correct answer is B (Subtract 5). Check: 20 − 5 = 15 ✓, 16 − 5 = 11 ✓, 30 − 5 = 25 ✓. A (Subtract 3) is wrong because 20 − 3 = 17, not 15. C (Divide by 2) is wrong because 20 ÷ 2 = 10, not 15. D (Subtract 10) is wrong because 20 − 10 = 10, not 15.

Question 4: APPLIED

Maria's lemonade stand sells cups of lemonade. She keeps track of how many lemons she uses. Cups made: 2 → Lemons used: 6 Cups made: 5 → Lemons used: 15 Cups made: 8 → Lemons used: 24 If Maria makes 12 cups, how many lemons will she use?

A. 24
B. 30
C. 36
D. 48

Show answer (C)

The correct answer is C (36). The rule is × 3 (each cup uses 3 lemons). Check: 2 × 3 = 6 ✓, 5 × 3 = 15 ✓, 8 × 3 = 24 ✓. So 12 × 3 = 36. A (24) would mean × 2, but 2 × 2 = 4, not 6. B (30) doesn't match any consistent rule. D (48) would mean × 4, but 2 × 4 = 8, not 6.

Question 5: CRITICAL THINKING

A number machine changes numbers using a secret rule. Input: 3 → Output: 15 Input: 5 → Output: 25 Input: 8 → Output: 40 Sam says the rule is "add 6." Is Sam correct, and what is the correct rule?

A. Sam is correct; the rule is add 6.
B. Sam is wrong; the rule is add 12.
C. Sam is wrong; the rule is multiply by 5.
D. Sam is wrong; the rule is multiply by 6.

Show answer (C)

The correct answer is C. Check Sam's rule (add 6) against every row: 3 + 6 = 9 ✓, but 5 + 6 = 11, not 15 ✗. Sam's rule fails, so Sam is wrong.Now check multiply by 5: 3 × 5 = 15 ✓, 5 × 5 = 25 ✓, 8 × 5 = 40 ✓. The rule multiply by 5 works for every row. Always check all rows before deciding on a rule!

SUMMARY

Let's Review!

Summary

An input-output table shows pairs of numbers connected by a hidden rule. To find the rule, look at the first row and ask: what operation (+, −, ×, or ÷) turns the input into the output? Then test your guess on every other row. If it works for all of them, you found the rule!

Quick clue: if the output is bigger, try addition or multiplication. If it's smaller, try subtraction or division. On the ISEE, use process of elimination — test each answer choice on the table rows. And remember, there's no penalty for guessing, so never leave a question blank. You've got this!